I've been taken by the concise result^{1}
that (roughly!), on a $2$-dimensional torus $\mathbb{T}^2$, the time it takes
to visit nearly every point (within $\epsilon$, as $\epsilon \to 0$) is: $\frac{2}{\pi}$.

My question is:

. Is the same situation known for the $d$-dimensional torus, $\mathbb{T}^d$? What is the time it takes for a Brownian-motion particle to visit within $\epsilon \to 0$ of every point of $\mathbb{T}^d$?Q

This is probably known—or known to be unknown—so this is just a reference request.

^{1}Dembo, Amir, Yuval Peres, Jay Rosen, and Ofer Zeitouni. "Cover times for Brownian motion and random walks in two dimensions."

*Annals of Mathematics*(2004): 433-464. Annals link.